(a) is obvious.
(b) From Theorem 1 we need to only show
that \([\eta _i]\)  is an infinite set if \(\eta _i\in T_1\) .
It is well-known from [1]} that \(u_1,u_2\in \mathbb {H} \)  are
conjugate (i.e., there exists nonzero \( q\in \mathbb {H}\) 
such that \(u_1=qu_2q^{-1}\) ) iff \(\operatorname{Re}u_1=\operatorname{Re}u_2\)  and
\(|u_1|=|u_2|\) . Since \(\eta _i\in T_1\)  has a
nonzero imaginary part, \([\eta _i]\)  is an infinite set. Thus \(q_nx^n+\cdots +q_1x+q_0=0\)  has a finite
number of roots in \(\mathbb {H}\)  iff \(T_1=\varnothing \) , iff \(p_nx^n+\cdots +p_1x+d_0=0\)  has
at most \(n\)  roots in \(\mathbb {H}\)  since \(s+k\le n\)  (See (REF ) for the notations).
(c) follows from \(s+k\le n\) .
